Dirichlet series
| L(s) = 1 | − 4·2-s − 5·3-s + 6·4-s + 4·5-s + 20·6-s + 4·7-s − 4·8-s + 19·9-s − 16·10-s − 2·11-s − 30·12-s + 8·13-s − 16·14-s − 20·15-s + 16-s − 13·17-s − 76·18-s + 15·19-s + 24·20-s − 20·21-s + 8·22-s + 20·23-s + 20·24-s + 6·25-s − 32·26-s − 55·27-s + 24·28-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 2.88·3-s + 3·4-s + 1.78·5-s + 8.16·6-s + 1.51·7-s − 1.41·8-s + 19/3·9-s − 5.05·10-s − 0.603·11-s − 8.66·12-s + 2.21·13-s − 4.27·14-s − 5.16·15-s + 1/4·16-s − 3.15·17-s − 17.9·18-s + 3.44·19-s + 5.36·20-s − 4.36·21-s + 1.70·22-s + 4.17·23-s + 4.08·24-s + 6/5·25-s − 6.27·26-s − 10.5·27-s + 4.53·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 7^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 7^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 5^{16} \cdot 7^{16} \cdot 11^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.17143\times 10^{12}\) |
| Root analytic conductor: | \(2.47961\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 5^{16} \cdot 7^{16} \cdot 11^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(5.836109384\) |
| \(L(\frac12)\) | \(\approx\) | \(5.836109384\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \) |
| 5 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \) | |
| 7 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \) | |
| 11 | \( 1 + 2 T + 40 T^{2} + 14 T^{3} + 651 T^{4} - 1088 T^{5} + 6178 T^{6} - 29354 T^{7} + 53113 T^{8} - 29354 p T^{9} + 6178 p^{2} T^{10} - 1088 p^{3} T^{11} + 651 p^{4} T^{12} + 14 p^{5} T^{13} + 40 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 3 | \( 1 + 5 T + 2 p T^{2} - 10 T^{3} - 25 T^{4} - 10 T^{5} - 19 p T^{6} - 110 T^{7} + 431 T^{8} + 1130 T^{9} - 128 p T^{10} - 2215 T^{11} + 1397 T^{12} + 40 T^{13} - 574 p^{3} T^{14} + 1120 p T^{15} + 65578 T^{16} + 1120 p^{2} T^{17} - 574 p^{5} T^{18} + 40 p^{3} T^{19} + 1397 p^{4} T^{20} - 2215 p^{5} T^{21} - 128 p^{7} T^{22} + 1130 p^{7} T^{23} + 431 p^{8} T^{24} - 110 p^{9} T^{25} - 19 p^{11} T^{26} - 10 p^{11} T^{27} - 25 p^{12} T^{28} - 10 p^{13} T^{29} + 2 p^{15} T^{30} + 5 p^{15} T^{31} + p^{16} T^{32} \) |
| 13 | \( 1 - 8 T + 12 p T^{3} - 30 T^{4} - 2804 T^{5} - 986 T^{6} + 53156 T^{7} - 2361 T^{8} - 672388 T^{9} - 199668 T^{10} + 6562678 T^{11} + 12990388 T^{12} - 64060416 T^{13} - 238206152 T^{14} + 36927560 p T^{15} + 2262051497 T^{16} + 36927560 p^{2} T^{17} - 238206152 p^{2} T^{18} - 64060416 p^{3} T^{19} + 12990388 p^{4} T^{20} + 6562678 p^{5} T^{21} - 199668 p^{6} T^{22} - 672388 p^{7} T^{23} - 2361 p^{8} T^{24} + 53156 p^{9} T^{25} - 986 p^{10} T^{26} - 2804 p^{11} T^{27} - 30 p^{12} T^{28} + 12 p^{14} T^{29} - 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 17 | \( 1 + 13 T + 64 T^{2} + 104 T^{3} - 257 T^{4} - 2 T^{5} + 17739 T^{6} + 131018 T^{7} + 614787 T^{8} + 2465284 T^{9} + 7672902 T^{10} + 16146399 T^{11} + 65097021 T^{12} + 602825528 T^{13} + 3635676786 T^{14} + 14260301756 T^{15} + 51779561314 T^{16} + 14260301756 p T^{17} + 3635676786 p^{2} T^{18} + 602825528 p^{3} T^{19} + 65097021 p^{4} T^{20} + 16146399 p^{5} T^{21} + 7672902 p^{6} T^{22} + 2465284 p^{7} T^{23} + 614787 p^{8} T^{24} + 131018 p^{9} T^{25} + 17739 p^{10} T^{26} - 2 p^{11} T^{27} - 257 p^{12} T^{28} + 104 p^{13} T^{29} + 64 p^{14} T^{30} + 13 p^{15} T^{31} + p^{16} T^{32} \) | |
| 19 | \( 1 - 15 T + 87 T^{2} - 275 T^{3} + 597 T^{4} - 1120 T^{5} + 2532 T^{6} + 21050 T^{7} - 358427 T^{8} + 3135015 T^{9} - 17730233 T^{10} + 59745165 T^{11} - 5330885 p T^{12} - 426457210 T^{13} + 3926054990 T^{14} - 13043437630 T^{15} + 41174254216 T^{16} - 13043437630 p T^{17} + 3926054990 p^{2} T^{18} - 426457210 p^{3} T^{19} - 5330885 p^{5} T^{20} + 59745165 p^{5} T^{21} - 17730233 p^{6} T^{22} + 3135015 p^{7} T^{23} - 358427 p^{8} T^{24} + 21050 p^{9} T^{25} + 2532 p^{10} T^{26} - 1120 p^{11} T^{27} + 597 p^{12} T^{28} - 275 p^{13} T^{29} + 87 p^{14} T^{30} - 15 p^{15} T^{31} + p^{16} T^{32} \) | |
| 23 | \( ( 1 - 10 T + 148 T^{2} - 1130 T^{3} + 9268 T^{4} - 59090 T^{5} + 350700 T^{6} - 1945810 T^{7} + 9369686 T^{8} - 1945810 p T^{9} + 350700 p^{2} T^{10} - 59090 p^{3} T^{11} + 9268 p^{4} T^{12} - 1130 p^{5} T^{13} + 148 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 29 | \( 1 + 14 T + 2 T^{2} - 1072 T^{3} - 6890 T^{4} + 4328 T^{5} + 9994 p T^{6} + 1781766 T^{7} + 1950555 T^{8} - 45933916 T^{9} - 350535774 T^{10} - 797309106 T^{11} + 5620165904 T^{12} + 53524798108 T^{13} + 121009884288 T^{14} - 709820470602 T^{15} - 6821549455803 T^{16} - 709820470602 p T^{17} + 121009884288 p^{2} T^{18} + 53524798108 p^{3} T^{19} + 5620165904 p^{4} T^{20} - 797309106 p^{5} T^{21} - 350535774 p^{6} T^{22} - 45933916 p^{7} T^{23} + 1950555 p^{8} T^{24} + 1781766 p^{9} T^{25} + 9994 p^{11} T^{26} + 4328 p^{11} T^{27} - 6890 p^{12} T^{28} - 1072 p^{13} T^{29} + 2 p^{14} T^{30} + 14 p^{15} T^{31} + p^{16} T^{32} \) | |
| 31 | \( 1 + 6 T + 50 T^{2} + 770 T^{3} + 5574 T^{4} + 52812 T^{5} + 413358 T^{6} + 2747730 T^{7} + 23419731 T^{8} + 155200888 T^{9} + 1038560710 T^{10} + 6987868918 T^{11} + 40971084744 T^{12} + 274659935542 T^{13} + 1610758333860 T^{14} + 8890068663118 T^{15} + 53194048809257 T^{16} + 8890068663118 p T^{17} + 1610758333860 p^{2} T^{18} + 274659935542 p^{3} T^{19} + 40971084744 p^{4} T^{20} + 6987868918 p^{5} T^{21} + 1038560710 p^{6} T^{22} + 155200888 p^{7} T^{23} + 23419731 p^{8} T^{24} + 2747730 p^{9} T^{25} + 413358 p^{10} T^{26} + 52812 p^{11} T^{27} + 5574 p^{12} T^{28} + 770 p^{13} T^{29} + 50 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 37 | \( 1 - 28 T + 248 T^{2} + 376 T^{3} - 20638 T^{4} + 92104 T^{5} + 672674 T^{6} - 7678558 T^{7} + 4366359 T^{8} + 274429580 T^{9} - 1018256312 T^{10} - 6667618492 T^{11} + 55772451208 T^{12} + 22346143252 T^{13} - 1395185549064 T^{14} + 404893621690 T^{15} + 40789850682361 T^{16} + 404893621690 p T^{17} - 1395185549064 p^{2} T^{18} + 22346143252 p^{3} T^{19} + 55772451208 p^{4} T^{20} - 6667618492 p^{5} T^{21} - 1018256312 p^{6} T^{22} + 274429580 p^{7} T^{23} + 4366359 p^{8} T^{24} - 7678558 p^{9} T^{25} + 672674 p^{10} T^{26} + 92104 p^{11} T^{27} - 20638 p^{12} T^{28} + 376 p^{13} T^{29} + 248 p^{14} T^{30} - 28 p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( 1 - 2 T - 56 T^{2} + 456 T^{3} + 4228 T^{4} - 540 T^{5} - 278272 T^{6} + 924298 T^{7} + 22816785 T^{8} - 26602182 T^{9} - 749035956 T^{10} + 1347482516 T^{11} + 56110313084 T^{12} + 18115550386 T^{13} - 2074406733188 T^{14} + 3193810608334 T^{15} + 97551488947641 T^{16} + 3193810608334 p T^{17} - 2074406733188 p^{2} T^{18} + 18115550386 p^{3} T^{19} + 56110313084 p^{4} T^{20} + 1347482516 p^{5} T^{21} - 749035956 p^{6} T^{22} - 26602182 p^{7} T^{23} + 22816785 p^{8} T^{24} + 924298 p^{9} T^{25} - 278272 p^{10} T^{26} - 540 p^{11} T^{27} + 4228 p^{12} T^{28} + 456 p^{13} T^{29} - 56 p^{14} T^{30} - 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( ( 1 + 5 T + 156 T^{2} + 1112 T^{3} + 14252 T^{4} + 2209 p T^{5} + 958199 T^{6} + 5367826 T^{7} + 47078316 T^{8} + 5367826 p T^{9} + 958199 p^{2} T^{10} + 2209 p^{4} T^{11} + 14252 p^{4} T^{12} + 1112 p^{5} T^{13} + 156 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 47 | \( 1 + 10 T + 90 T^{2} + 990 T^{3} + 10962 T^{4} + 125780 T^{5} + 1049940 T^{6} + 8694110 T^{7} + 78182813 T^{8} + 657699020 T^{9} + 5516133590 T^{10} + 41660136620 T^{11} + 314453228334 T^{12} + 50762198450 p T^{13} + 17414600985600 T^{14} + 123891211917050 T^{15} + 853844816186055 T^{16} + 123891211917050 p T^{17} + 17414600985600 p^{2} T^{18} + 50762198450 p^{4} T^{19} + 314453228334 p^{4} T^{20} + 41660136620 p^{5} T^{21} + 5516133590 p^{6} T^{22} + 657699020 p^{7} T^{23} + 78182813 p^{8} T^{24} + 8694110 p^{9} T^{25} + 1049940 p^{10} T^{26} + 125780 p^{11} T^{27} + 10962 p^{12} T^{28} + 990 p^{13} T^{29} + 90 p^{14} T^{30} + 10 p^{15} T^{31} + p^{16} T^{32} \) | |
| 53 | \( 1 + 2 T - 90 T^{2} - 114 T^{3} + 1766 T^{4} - 39260 T^{5} + 84698 T^{6} + 3479298 T^{7} + 2505659 T^{8} + 26108952 T^{9} + 328127090 T^{10} - 9562142350 T^{11} - 56782562676 T^{12} + 46687386702 T^{13} - 955911781548 T^{14} + 11354489119094 T^{15} + 256942601322361 T^{16} + 11354489119094 p T^{17} - 955911781548 p^{2} T^{18} + 46687386702 p^{3} T^{19} - 56782562676 p^{4} T^{20} - 9562142350 p^{5} T^{21} + 328127090 p^{6} T^{22} + 26108952 p^{7} T^{23} + 2505659 p^{8} T^{24} + 3479298 p^{9} T^{25} + 84698 p^{10} T^{26} - 39260 p^{11} T^{27} + 1766 p^{12} T^{28} - 114 p^{13} T^{29} - 90 p^{14} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 59 | \( 1 - 7 T - 131 T^{2} + 1035 T^{3} + 4505 T^{4} - 103096 T^{5} + 580362 T^{6} + 5934746 T^{7} - 78352685 T^{8} + 42184295 T^{9} + 4051231229 T^{10} - 37639854343 T^{11} - 6067203859 T^{12} + 3027645888400 T^{13} - 16850302743700 T^{14} - 82397687707366 T^{15} + 1439734785923212 T^{16} - 82397687707366 p T^{17} - 16850302743700 p^{2} T^{18} + 3027645888400 p^{3} T^{19} - 6067203859 p^{4} T^{20} - 37639854343 p^{5} T^{21} + 4051231229 p^{6} T^{22} + 42184295 p^{7} T^{23} - 78352685 p^{8} T^{24} + 5934746 p^{9} T^{25} + 580362 p^{10} T^{26} - 103096 p^{11} T^{27} + 4505 p^{12} T^{28} + 1035 p^{13} T^{29} - 131 p^{14} T^{30} - 7 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( 1 - 4 T - 28 T^{2} - 428 T^{3} + 8866 T^{4} - 33948 T^{5} - 58628 T^{6} - 2463236 T^{7} + 35309927 T^{8} - 131877388 T^{9} - 370686656 T^{10} - 4142287130 T^{11} + 1814174192 p T^{12} - 118363478944 T^{13} - 4638563311768 T^{14} - 1369147620912 T^{15} + 335568946663249 T^{16} - 1369147620912 p T^{17} - 4638563311768 p^{2} T^{18} - 118363478944 p^{3} T^{19} + 1814174192 p^{5} T^{20} - 4142287130 p^{5} T^{21} - 370686656 p^{6} T^{22} - 131877388 p^{7} T^{23} + 35309927 p^{8} T^{24} - 2463236 p^{9} T^{25} - 58628 p^{10} T^{26} - 33948 p^{11} T^{27} + 8866 p^{12} T^{28} - 428 p^{13} T^{29} - 28 p^{14} T^{30} - 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 67 | \( ( 1 - 33 T + 842 T^{2} - 15042 T^{3} + 229138 T^{4} - 2884273 T^{5} + 32190375 T^{6} - 311931366 T^{7} + 2722654900 T^{8} - 311931366 p T^{9} + 32190375 p^{2} T^{10} - 2884273 p^{3} T^{11} + 229138 p^{4} T^{12} - 15042 p^{5} T^{13} + 842 p^{6} T^{14} - 33 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 71 | \( 1 - 2 T - 84 T^{2} + 1162 T^{3} + 3920 T^{4} - 44064 T^{5} + 367118 T^{6} + 427994 T^{7} + 9987787 T^{8} + 313362360 T^{9} + 1837418796 T^{10} - 6707925850 T^{11} - 70615169826 T^{12} + 3541571592582 T^{13} + 6280266794960 T^{14} - 63192636388642 T^{15} + 1577007356502727 T^{16} - 63192636388642 p T^{17} + 6280266794960 p^{2} T^{18} + 3541571592582 p^{3} T^{19} - 70615169826 p^{4} T^{20} - 6707925850 p^{5} T^{21} + 1837418796 p^{6} T^{22} + 313362360 p^{7} T^{23} + 9987787 p^{8} T^{24} + 427994 p^{9} T^{25} + 367118 p^{10} T^{26} - 44064 p^{11} T^{27} + 3920 p^{12} T^{28} + 1162 p^{13} T^{29} - 84 p^{14} T^{30} - 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 73 | \( 1 - 12 T - 70 T^{2} + 894 T^{3} - 3174 T^{4} + 15680 T^{5} + 979258 T^{6} - 6106968 T^{7} - 35840321 T^{8} + 219088858 T^{9} - 2250230130 T^{10} + 16042345080 T^{11} + 174867012124 T^{12} - 3122403626152 T^{13} + 5603105034612 T^{14} + 146703069126016 T^{15} - 1228549790232479 T^{16} + 146703069126016 p T^{17} + 5603105034612 p^{2} T^{18} - 3122403626152 p^{3} T^{19} + 174867012124 p^{4} T^{20} + 16042345080 p^{5} T^{21} - 2250230130 p^{6} T^{22} + 219088858 p^{7} T^{23} - 35840321 p^{8} T^{24} - 6106968 p^{9} T^{25} + 979258 p^{10} T^{26} + 15680 p^{11} T^{27} - 3174 p^{12} T^{28} + 894 p^{13} T^{29} - 70 p^{14} T^{30} - 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 79 | \( 1 - 2 T - 248 T^{2} - 554 T^{3} + 34544 T^{4} + 136224 T^{5} - 3433286 T^{6} - 7617606 T^{7} + 314192731 T^{8} + 96819992 T^{9} - 36339375384 T^{10} - 632139902 T^{11} + 3411895664398 T^{12} + 3442148324898 T^{13} - 242231927008752 T^{14} - 264097536737906 T^{15} + 17886559079534183 T^{16} - 264097536737906 p T^{17} - 242231927008752 p^{2} T^{18} + 3442148324898 p^{3} T^{19} + 3411895664398 p^{4} T^{20} - 632139902 p^{5} T^{21} - 36339375384 p^{6} T^{22} + 96819992 p^{7} T^{23} + 314192731 p^{8} T^{24} - 7617606 p^{9} T^{25} - 3433286 p^{10} T^{26} + 136224 p^{11} T^{27} + 34544 p^{12} T^{28} - 554 p^{13} T^{29} - 248 p^{14} T^{30} - 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 83 | \( 1 + 5 T - 167 T^{2} - 3187 T^{3} - 1213 T^{4} + 371522 T^{5} + 4214476 T^{6} + 7620588 T^{7} - 304541303 T^{8} - 4827248283 T^{9} - 31516312461 T^{10} + 119228514801 T^{11} + 4562805883611 T^{12} + 44293258449128 T^{13} + 85720816221504 T^{14} - 2646124131963366 T^{15} - 39705920909647216 T^{16} - 2646124131963366 p T^{17} + 85720816221504 p^{2} T^{18} + 44293258449128 p^{3} T^{19} + 4562805883611 p^{4} T^{20} + 119228514801 p^{5} T^{21} - 31516312461 p^{6} T^{22} - 4827248283 p^{7} T^{23} - 304541303 p^{8} T^{24} + 7620588 p^{9} T^{25} + 4214476 p^{10} T^{26} + 371522 p^{11} T^{27} - 1213 p^{12} T^{28} - 3187 p^{13} T^{29} - 167 p^{14} T^{30} + 5 p^{15} T^{31} + p^{16} T^{32} \) | |
| 89 | \( ( 1 - T + 504 T^{2} - 1162 T^{3} + 117472 T^{4} - 392061 T^{5} + 17106935 T^{6} - 63616750 T^{7} + 1769275296 T^{8} - 63616750 p T^{9} + 17106935 p^{2} T^{10} - 392061 p^{3} T^{11} + 117472 p^{4} T^{12} - 1162 p^{5} T^{13} + 504 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( 1 - 22 T - 50 T^{2} + 5896 T^{3} - 56535 T^{4} - 139482 T^{5} + 7896814 T^{6} - 89152106 T^{7} + 340506787 T^{8} + 5971323672 T^{9} - 115976796742 T^{10} + 727607121004 T^{11} + 3780959954979 T^{12} - 98901716769178 T^{13} + 498040115442938 T^{14} + 3252035542155752 T^{15} - 65108704374482496 T^{16} + 3252035542155752 p T^{17} + 498040115442938 p^{2} T^{18} - 98901716769178 p^{3} T^{19} + 3780959954979 p^{4} T^{20} + 727607121004 p^{5} T^{21} - 115976796742 p^{6} T^{22} + 5971323672 p^{7} T^{23} + 340506787 p^{8} T^{24} - 89152106 p^{9} T^{25} + 7896814 p^{10} T^{26} - 139482 p^{11} T^{27} - 56535 p^{12} T^{28} + 5896 p^{13} T^{29} - 50 p^{14} T^{30} - 22 p^{15} T^{31} + p^{16} T^{32} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.63397794976757440853323025854, −2.46847932342968312964028477367, −2.41642179482555713401762499222, −2.27902543856462347780028926469, −2.24704491706243020660887805469, −2.22523913709542776026533992556, −2.16729072844583671602957050666, −2.15107040822125609276185906769, −1.91292914718428032621642326260, −1.82721945723890903553457146202, −1.81212717959815605660375564155, −1.64862663543279525337608781503, −1.56094933278138180448642035520, −1.53653739108378959345099883193, −1.39526451869873737992727958027, −1.29778826921796046356221761234, −1.16428656433538335853358396575, −1.06145668071880319308554622809, −0.915944889046582870091855488884, −0.78701979824071185743574844746, −0.75720694913220031175536276100, −0.61898231254880220344717827039, −0.56142577142602890964145999971, −0.49777201931125311972629229647, −0.38203433488891065566875038475, 0.38203433488891065566875038475, 0.49777201931125311972629229647, 0.56142577142602890964145999971, 0.61898231254880220344717827039, 0.75720694913220031175536276100, 0.78701979824071185743574844746, 0.915944889046582870091855488884, 1.06145668071880319308554622809, 1.16428656433538335853358396575, 1.29778826921796046356221761234, 1.39526451869873737992727958027, 1.53653739108378959345099883193, 1.56094933278138180448642035520, 1.64862663543279525337608781503, 1.81212717959815605660375564155, 1.82721945723890903553457146202, 1.91292914718428032621642326260, 2.15107040822125609276185906769, 2.16729072844583671602957050666, 2.22523913709542776026533992556, 2.24704491706243020660887805469, 2.27902543856462347780028926469, 2.41642179482555713401762499222, 2.46847932342968312964028477367, 2.63397794976757440853323025854
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.